I'm not a native so this is just to start the ball rolling.
I think put oneself together (again) is merely an altered version of pull oneself together.Hence they mean basically the same.
However, it seems that put oneself is more used in a physical sense (clothes, hair,....) and pull in a psychological sense.

I can imagine that there could be instances when someone might say "I need to put myself back together" or I need "to put myself together". I would understand the metaphor and it wouldn't sound "wrong," though for me this isn't a common expression.

I think it would have a slightly different meaning from pull oneself together.

Pull oneself together implies that you are using your internal resources and fortitude to reconstitute yourself in a fairly sudden manner.

Put oneself together seems more like reconstructing yourself piece by piece. Sometimes we say, put the pieces back together, after some life-shattering event:

After my mother died, I had to gradually put the pieces of my life back together...

I think one could also say:

After my mother died, I had to put myself back together bit by bit.

Also, more colloquially (means the same as pull yourself together):

You are a mess. Get yourself together.You are a mess. Get it together.

i.e. He succeeded by using only his own resources. Used (only BE?) for people who start from a lowly background and rise up to be rich and famous.

Of course there is the obviously literal meaning, "He fell down and pulled himself up by grabbing a branch."

Click to expand...

Another possible use of 'pull oneself up' is when one suddenly realises one is about to blunder socially and stops oneself saying or doing something inappropriate. For example:
"He was about to tell his pub joke to Mary when he pulled himself up, remembering that she had just lost her brother to alcoholism."
I think 'pulling up' basically refers to restraining horses and bringing them to a standstill.